Tight Bounds on the Asymptotic Descriptive Complexity of Subgraph Isomorphism
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Let v(F) denote the number of vertices in a fixed connected pattern graph F. We show an infinite family of patterns F such that the existence of a subgraph isomorphic to F is expressible by a first-order sentence of quantifier depth 2/3 v(F)+1, assuming that the host graph is sufficiently large and connected. On the other hand, this is impossible for any F with using less than 2/3 v(F)-2 first-order variables.
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